Abstract
The Nakamura number of a simple game plays a critical role in preference aggregation (or multi-criterion ranking): the number of alternatives that the players can always deal with rationally is less than this number. We comprehensively study the restrictions that various properties for a simple game impose on its Nakamura number. We find that a computable game has a finite Nakamura number greater than three only if it is proper, nonstrong, and nonweak, regardless of whether it is monotonic or whether it has a finite carrier. The lack of strongness often results in alternatives that cannot be strictly ranked.
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We would like to thank an anonymous referee for useful suggestions. The discussion in footnote 3 and Remark 4, among other things, would not have been possible without his/her suggestion.
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Kumabe, M., Mihara, H.R. The Nakamura numbers for computable simple games. Soc Choice Welfare 31, 621–640 (2008). https://doi.org/10.1007/s00355-008-0300-5
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DOI: https://doi.org/10.1007/s00355-008-0300-5